A snapshot of the theory of orthogonal polynomials on the line or on the circle is presented from the point of view of the asymptotics of bounded point evaluation constants, touching on classical fields of approximation theory of real functions, as well as complex analytic functions of a single variable.

The stability of the Christoffel-Darboux kernel under small perturbations of the generating measure is established via precise quantitative bounds. Trace-class perturbations of the Hessenberg matrix attached to a 2D measure are linked to the asymptotic invariance of the Christoffel function, in an exact separation algorithm of outliers from clouds formed by bounded point evaluations for complex analytic functions.

Starting from the explicit formulas known for simple multivariate geometries (balls, cube, simplex) the discussion moves to recent advances in pluripotential theory. The essential Bernstein-Markov property of a measure is necessary for deriving asymptotics of the Christoffel function outside the support of the generating measure, and in some fortunate situations also inside the support. A balance between strong and weak limits enters into the game.

Several applicationsare described of the Christoffel-Darboux kernel in computational statistics, including parametric (polynomial) regression, optimal design (and an interpretation in computational geometry), density approximation, support inference and outlier detection. Theoretical results leverage statistical concentration and properties of the Christoffel-Darboux kernel. They are illustrated with numerical experiments.

Outside the immediate statistical applications, two notable implications of the Christoffel-Darboux kernel are sketched: on the effective semialgebraic approximation of nonsmooth functions and on the spectral analysis of Koopman's operator attached to some intricate dynamical systems.

A spectral characterization of the Christoffel function is provided via its associated moment matrix, and that of Dirac measures. A first (Hahn-Banach) extension of the Christoffel function to Lp spaces is characterized. A second extension is also provided and analyzed where squares of polynomials (hence positive everywhere) are replaced by larger convex cones of polynomials positive on the support of the underlying measure, in particular polynomials in a certain associated quadratic module.

A general overview of the book is detailed after a brief description of the main algorithm to compute and exploit Christoffel-Darboux from raw moment data.

Basic concepts and results of real and complex algebraic geometry enter into play when dealing with the estimation of measures supported by algebraic varieties. Some geometric constraints on the supporting variety are inherited from the very beginning of the modelization, such as a sphere for estimating an orientation in space. Well-adapted results of multivariate pluripotential theory play a crucial role here.

A brief introduction based on several classical examples to the correspondence between positive-definite kernels and Hilbert spaces of functions is tuned around the Christoffel-Darboux kernel and its relevance to moment problems.